What is the equation of motion for forced vibrations?

Short Answer:

The equation of motion for forced vibrations describes the relationship between the external force acting on a vibrating system and the system’s displacement, mass, stiffness, and damping. It helps to understand how the system behaves when subjected to periodic external forces.

In simple words, the equation shows that the total external force is equal to the sum of the inertia force, damping force, and restoring force. It is written as

where  is mass,  is damping coefficient,  is stiffness, and  is the external periodic force.

Detailed Explanation :

Equation of Motion for Forced Vibrations

Forced vibrations occur when an external time-varying force acts on a mechanical system and causes it to vibrate continuously at the frequency of that external force. The equation of motion for forced vibration expresses the dynamic equilibrium between the external force and the forces developed in the system, namely the inertia force, damping force, and restoring force.

To understand this concept clearly, let us consider a simple spring–mass–damper system subjected to an external periodic force.

Derivation of the Equation of Motion

Let:

  •  = mass of the vibrating body (kg)
  •  = damping coefficient (N·s/m)
  •  = stiffness of the spring (N/m)
  •  = displacement of the mass at any time  (m)
  •  = externally applied force =  (N)

When the system vibrates under the action of this external force, the following three forces act on it:

  1. Inertia Force (Opposing Acceleration):
    This is the force required to accelerate the mass. It is proportional to the acceleration of the mass and acts in the opposite direction.
  1. Damping Force (Opposing Velocity):
    This is the resistive force due to damping. It is proportional to the velocity of the mass and also acts in the opposite direction to motion.
  1. Restoring Force (Opposing Displacement):
    This is the elastic force developed in the spring, which is proportional to the displacement of the mass. It always acts opposite to the direction of displacement.

According to Newton’s Second Law of Motion, the algebraic sum of all the forces acting on the mass must be equal to the external force applied:

Substituting the individual forces:

Rearranging the terms gives the equation of motion for forced vibration as:

This is the standard differential equation representing a single-degree-of-freedom system subjected to a harmonic external force.

Explanation of Each Term

  1. :
    This term represents the inertia force. It is proportional to the acceleration and resists changes in motion.
  2. :
    This term represents the damping force, which dissipates energy and controls the amplitude of vibration.
  3. :
    This is the restoring force due to the spring, which tries to bring the mass back to its equilibrium position.
  4. :
    This term represents the periodic external force that keeps the system vibrating.

The equation shows that the external force is balanced by the three internal forces acting within the system.

Special Cases of the Equation

The equation of motion can take different forms depending on whether damping is present or not.

  1. Without Damping (c = 0):
    For an undamped system, the equation becomes:

This represents an ideal system with no energy loss, where vibrations can continue indefinitely.

  1. With Damping (c ≠ 0):
    For a damped system, energy is continuously lost due to damping, and the amplitude remains limited even at resonance. The general equation remains:

Solution of the Equation

The complete solution of the equation consists of two parts:

  1. Transient Solution (x₁):
    This part represents the natural vibration of the system, which gradually dies out over time due to damping.
  2. Steady-State Solution (x₂):
    This part represents the forced vibration that continues as long as the external force acts.

Thus,

where  is the transient component and  is the steady-state component.

In practice, the transient vibrations disappear after a few cycles, and the system continues to vibrate in the steady-state mode at the frequency of the external force.

Amplitude of Forced Vibration

From the steady-state solution, the amplitude  of vibration can be expressed as:

Where:

  •  = frequency ratio
  •  = damping ratio

This expression shows how the amplitude depends on the frequency of the external force and the amount of damping in the system.

Significance of the Equation

The equation of motion for forced vibration helps engineers to:

  • Predict how systems will respond to external forces.
  • Determine the vibration amplitude, phase, and frequency.
  • Design mechanical systems to avoid resonance.
  • Analyze the effect of damping on vibration control.

It forms the foundation of vibration analysis and is widely used in designing machines, vehicles, buildings, and structures subjected to dynamic loads.

Conclusion

In conclusion, the equation of motion for forced vibrations is

It shows that the external periodic force applied to a system is balanced by the inertia, damping, and restoring forces within the system. This equation helps engineers understand and control the dynamic behavior of machines and structures under vibration. By adjusting the damping and stiffness, the effects of resonance and excessive vibration can be minimized, ensuring safe and stable operation.